Central automorphisms of Laguerre planes

Any automorphism of a Benz plane having at least one fixed point induces a collineation on the projective extension of the residual affine plane with reference to this point. When this collineation is a central automorphism , the initial automorphism is called the central automorphism (or central-axial automorphism, cf.([3]). In this paper we present an analytical description of central automorphisms of a miguelian Laguerre planes with the characteristic different from two. This description is applied to find transitive groups of homotheties and translations of types occuring in the classification theorems of R. Kleinewillinghofer ([2]). Some examples over an arbitrary commutative field are constructed, the other over the finite field Z3 and z5. It is interesting that two types of the Kleinewillinghofer classification ( [2] ) appear only as automorphism subgroups of finite plane of order three or five. This will give a clear characterization of these planes. Throughaut we assume that the characteristic of a plane is not equal to two.